Binary ↘ Double: The 64 Bit Double Precision IEEE 754 Binary Floating Point Standard Representation Number 0 - 010 0000 0000 - 1100 1110 0000 0000 0000 0000 0000 0000 0000 0000 0000 0011 0100 Converted and Written as a Base Ten Decimal System Number (as a Double)
0 - 010 0000 0000 - 1100 1110 0000 0000 0000 0000 0000 0000 0000 0000 0000 0011 0100: 64 bit double precision IEEE 754 binary floating point standard representation number converted to decimal system (base ten)
1. Identify the elements that make up the binary representation of the number:
The first bit (the leftmost) indicates the sign,
1 = negative, 0 = positive.
0
The next 11 bits contain the exponent:
010 0000 0000
The last 52 bits contain the mantissa:
1100 1110 0000 0000 0000 0000 0000 0000 0000 0000 0000 0011 0100
2. Convert the exponent from binary (from base 2) to decimal (in base 10).
The exponent is allways a positive integer.
010 0000 0000(2) =
0 × 210 + 1 × 29 + 0 × 28 + 0 × 27 + 0 × 26 + 0 × 25 + 0 × 24 + 0 × 23 + 0 × 22 + 0 × 21 + 0 × 20 =
0 + 512 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 =
512 =
512(10)
3. Adjust the exponent.
Subtract the excess bits: 2(11 - 1) - 1 = 1023,
that is due to the 11 bit excess/bias notation.
The exponent, adjusted = 512 - 1023 = -511
4. Convert the mantissa from binary (from base 2) to decimal (in base 10).
The mantissa represents the fractional part of the number (what comes after the whole part of the number, separated from it by a comma).
1100 1110 0000 0000 0000 0000 0000 0000 0000 0000 0000 0011 0100(2) =
1 × 2-1 + 1 × 2-2 + 0 × 2-3 + 0 × 2-4 + 1 × 2-5 + 1 × 2-6 + 1 × 2-7 + 0 × 2-8 + 0 × 2-9 + 0 × 2-10 + 0 × 2-11 + 0 × 2-12 + 0 × 2-13 + 0 × 2-14 + 0 × 2-15 + 0 × 2-16 + 0 × 2-17 + 0 × 2-18 + 0 × 2-19 + 0 × 2-20 + 0 × 2-21 + 0 × 2-22 + 0 × 2-23 + 0 × 2-24 + 0 × 2-25 + 0 × 2-26 + 0 × 2-27 + 0 × 2-28 + 0 × 2-29 + 0 × 2-30 + 0 × 2-31 + 0 × 2-32 + 0 × 2-33 + 0 × 2-34 + 0 × 2-35 + 0 × 2-36 + 0 × 2-37 + 0 × 2-38 + 0 × 2-39 + 0 × 2-40 + 0 × 2-41 + 0 × 2-42 + 0 × 2-43 + 0 × 2-44 + 0 × 2-45 + 0 × 2-46 + 1 × 2-47 + 1 × 2-48 + 0 × 2-49 + 1 × 2-50 + 0 × 2-51 + 0 × 2-52 =
0.5 + 0.25 + 0 + 0 + 0.031 25 + 0.015 625 + 0.007 812 5 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0.000 000 000 000 007 105 427 357 601 001 858 711 242 675 781 25 + 0.000 000 000 000 003 552 713 678 800 500 929 355 621 337 890 625 + 0 + 0.000 000 000 000 000 888 178 419 700 125 232 338 905 334 472 656 25 + 0 + 0 =
0.5 + 0.25 + 0.031 25 + 0.015 625 + 0.007 812 5 + 0.000 000 000 000 007 105 427 357 601 001 858 711 242 675 781 25 + 0.000 000 000 000 003 552 713 678 800 500 929 355 621 337 890 625 + 0.000 000 000 000 000 888 178 419 700 125 232 338 905 334 472 656 25 =
0.804 687 500 000 011 546 319 456 101 628 020 405 769 348 144 531 25(10)
5. Put all the numbers into expression to calculate the double precision floating point decimal value:
(-1)Sign × (1 + Mantissa) × 2(Adjusted exponent) =
(-1)0 × (1 + 0.804 687 500 000 011 546 319 456 101 628 020 405 769 348 144 531 25) × 2-511 =
1.804 687 500 000 011 546 319 456 101 628 020 405 769 348 144 531 25 × 2-511 =
0.000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 269 199 485 766 759 184 468 352 177 198 785 374 322 118 671 473 466 611 070 610 156 493 352 462 490 303 833 332 618 602 316 294 847 479 767 032 593 247 711 896 004 399 719 740 398 746 002 701 338 137 221 1
0 - 010 0000 0000 - 1100 1110 0000 0000 0000 0000 0000 0000 0000 0000 0000 0011 0100 converted from a 64 bit double precision IEEE 754 binary floating point standard representation number to a decimal system number, written in base ten (double) = 0.000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 269 199 485 766 759 184 468 352 177 198 785 374 322 118 671 473 466 611 070 610 156 493 352 462 490 303 833 332 618 602 316 294 847 479 767 032 593 247 711 896 004 399 719 740 398 746 002 701 338 137 221 1(10)
Spaces were used to group digits: for binary, by 4, for decimal, by 3.
More operations with 64 bit double precision IEEE 754 binary floating point standard representation numbers: